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André–Quillen cohomology
In commutative algebra, André–Quillen cohomology is a theory of cohomology for commutative rings which is closely related to the cotangent complex. The first three cohomology groups were introduced by and are sometimes called Lichtenbaum–Schlessinger functors ''T''0, ''T''1, ''T''2, and the higher groups were defined independently by Michel André and by Daniel Quillen using methods of homotopy theory. It comes with a parallel homology theory called André–Quillen homology.
==Motivation==
Let ''A'' be a commutative ring, ''B'' be an ''A''-algebra, and ''M'' be a ''B''-module. André–Quillen cohomology is the derived functors of the derivation functor Der''A''(''B'', ''M''). Before the general definitions of André and Quillen, it was known for a long time that given morphisms of commutative rings and a ''C''-module ''M'', there is a three-term exact sequence of derivation modules:
:
This term can be extended to a six-term exact sequence using the functor Exalcomm of extensions of commutative algebras and a nine-term exact sequence using the Lichtenbaum–Schlessinger functors. André–Quillen cohomology extends this exact sequence even further. In the zeroth degree, it is the module of derivations; in the first degree, it is Exalcomm; and in the second degree, it is the second degree Lichtenbaum–Schlessinger functor.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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